In this section, by filtration of a chain complex (respectively, of a graded module) \(C_*\)  we mean a collection \(\lbrace  F_p C_* \rbrace _{p\in \mathbb {Z}}\)  of chain subcomplexes (resp. submodules) such that \(F_{p-1} C_* \subseteq F_p C_*\) , for all \(p\in \mathbb {Z}\) . A filtration \(\lbrace  F_p C_* \rbrace _{p\in \mathbb {Z}}\)  is said to be bounded if, for every \(k\) , there exist integers \(s(k)\)  and \(t(k)\)  such that \(F_{s(k)}C_k =0\)  and \(F_{t(k)}C_k = C_k\) .
We can use bounded filtrations to introduce the notion of convergence of a spectral sequence. A more general notion of convergence can be defined [1]}, but it is not necessary for the purpose of this work.
A spectral sequence \(\lbrace E^r_{p,q}, d^r_{p,q}\rbrace \)  is said to converge to a graded module \(H_*\)  if there is a bounded filtration \(\lbrace  \mathcal {F}_p H_* \rbrace _{p\in \mathbb {Z}}\)  of \(H_*\)  such that
\(E^{\infty }_{p,q} \cong \mathcal {F}_p H_{k} / \mathcal {F}_{p-1} H_{k}\) , for all \(p\) , \(q\)  and \(k=p+q\) . In this paper, we adopt a typical convention for the indices of a spectral sequence: whenever \(p\) , \(q\)  and \(k\)  appear together, they are related by \(k=p+q\) . Notice that, since we are considering vector spaces, convergence can be rephrased as
\(H_k \cong \bigoplus _{p+q=k} E^{\infty }_{p,q} ;\) 
